Lectures on Image Processing

YW EECE 4353 Image Processing

Vanderbilt University School of Engineering

EECE/CS 4353 Image Processing

Lecture Notes: Image Sharpening

Richard Alan Peters II

Department of Electrical and Computer Engineering
Fall Semester 2021

YW EECE 4353 Image Processing

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Sharpening

e Results from high frequency enhancement since
small features correspond to short wavelength
sinusoids.

e Relative amplification of high frequencies in the
Fourier domain corresponds to differentiation in
the spatial domain.

e Ona discrete image, differentiation corresponds
to pixel differencing.

11 October 2021 ©1999-2021 by Richard Alan Peters IT 2

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The Derivative Property of the Fourier Transform

The FT of the partial
derivativew.r.t.r(in J Ol
the row direction) of

i
{os
,—os

Za(re)ee” ded

Integration

an image, L ... os by parts

=f j U(r,c)- 2 enue) dedr

= J j I(r,c) Cine Prnen dedr

wexoery -i22(ue+vr)
.. is equal to the product of =e J J Ur.e)e ddr
the FT of the image and the ail This results in
corresponding frequency = -i2av { I} =-i2zvF(u,v). | horizontal HF
variable, v. enhancement

11 October 2021 ©1999-2021 by Richard Alan Peters IT 3

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Differentiation is Highpass Filtering

Vertical HF

Enhancement |
5 u,v) a uF{T1}(w,v)
5 u,v) 0 vE{T} (u,v

Directional Horizontal HF
derivative in r. Enhancement

Directional
derivative in c.

=—_— oo
—_~ —™

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Fourier Transforms of Sums of Derivatives

a{[2-2)h} =~i2n{u+v) 81} =-i2(u+v)F (uv).

or ac

Sum of first-order wlinear amplification
partial derivatives... of high frequencies

or ace

a[S-Sh =—42°(u? +v") 5{1} =-42°(u?+v°)F(uy).

Sum of second-order quadratic amplification
partial derivatives... of high frequencies

11 October 2021 ©1999-2021 by Richard Alan Peters IT 5

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Sharpening via Differencing or Highpass Filtering

Sharpening results from adding to the image a
copy of itself that has been:

e Pixel-differenced in the spatial domain:

- Each pixel in the output is a difference between itself
and a weighted average of its neighbors.

- Is aconvolution whose weight matrix sums to 0.
e Highpass filtered in the frequency domain:
- High frequencies are enhanced or amplified.

- Individual frequency components are multiplied by an
increasing function of w such as aw = @V(u?+v), where
a is a constant.

11 October 2021 ©1999-2021 by Richard Alan Peters IT 6

Horizontal Differences

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Image

fac

Left (back)
Difference

Right (fwd)
Difference

Sum of
Differences

Foe li

a

| W255

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Horizontal Differencing / Sharpening

original: I(r,c) upward diff: I(r,c)-I(r-1,c) sharpened: 2I(r,c)-I(r-1,c)

11 October 2021 ©1999-2021 by Richard Alan Peters IT 8

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Horizontal Differencing / Sharpening

original: I(r,c) downward diff: I(r,c)-I(r+1,c) sharpened: 21(r,c)-I(r+1,c)

11 October 2021 ©1999-2021 by Richard Alan Peters IT 9

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Horizontal Differencing / Sharpening

original: I(r,c) 21(r,c)-I(-1,c)-I(r+ 1c) 31(r,0)-I0r-1,0)-I(r+ 1,0)

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Vertical Differencing / Sharpening

original: I(r,c) backward diff: I(r,c)-I(r,c-1) sharpened: 2I(r,c)-I(r,c-1)

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Vertical Differencing / Sharpening

original: I(r,c) forward diff: I(r,c)-I(r,c+1) sharpened: 21(r,c)-[(r,ct+1)

11 October 2021 ©1999-2021 by Richard Alan Peters II 12

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Vertical Sharpening

original: I(r,c) 21(r,c)-I(r,c-1)-I(r,c+1), 31(r,c)-I(r,c-1)-I(r,c+1),

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Horizontal + Vertical Sharpening

original: I(r,c) 41(r,0)-Wryct1)-I(r,c-1)- S1(r,0)-Mr,e+1)-I(r,c-1)-
I(r+/,c)-I(r-1,0) I(r+1,c)-I(-1,c)

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That our visual system does something like the edge
enhancement of the disk on the left is strongly

Perceptual Note suggested by the appearance of the disk on the
right. It contains only 2 intensity levels. But, we see
4 - the background, the disk, and concentric dark
and bright circles surrounding the disk.

51(r,c)-I(r,c+1)-I(r,c-1)- a two-level image:
U(rt1c)-1-1,c) I(r,c) € {64,192}

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Differencing / Highpass Filtering

original image, I power spectrum

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Differencing / Highpass Filtering

power spectrum of h = [-1 1 0] power spectrum of I*h = I(r,c)-I(r,c+1)

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Differencing / Highpass Filtering

negative pixels in differenced image positive pixels in differenced image

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Differencing / Highpass Filtering

original image, I power spectrum

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¥7 Add the differenced image, EECE 4353 Image Processing
I(r,c)-I(r,c+1), back to the Vanderbilt University School of Engineering

original to get a HF enhanced
version. It is a “sharper”
version of the original.

ening

sharpened image, 2I(r,c)-I(r,c+1) power spectrum

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Image Sharpening

original image, I power spectrum

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Image Sharpening

shift to the right of
the sharpened image.

sharpened image, 2I(r,c)-I(r,c+1) power spectrum

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Image Sharpening: E

Blured Ege (Pie) + Forward Difoence

dge Enhancement

Blurred Edge (rop) + Forward Dflerence

stepedge location

step ede location

ims ty edge
foxyatcetterence

sum of edge and ference

Adding a differenced image back to the original

increases the high frequency content. It

steepens the slopes of the edges which makes the image look “sharper.” Note also that a

forward difference, I(r,c)-I(r,c+1), causes the ap)

arent edge to shift to the right.

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Image Sharpening: Edge Enhancement

lured Edge (Ris) + Backward Oference

Bleed Edge (Drop) + Backward Dilorence

stepedge location

step edge location

Adding a differenced image back to the original increases the high frequency content. It
steepens the slopes of the edges which makes the image look “sharper.” Note also that a
backward difference, I(r,c)-I(r,c-1), causes the apparent edge to shift to the left.

11 October 2021

©1999-2021 by Richard Alan Peters IT 24

YW The shift occurs because the EECE 4353 Image Processing
direction of the differencing Vanderbilt University School of Engineering

operation pushes edges in the
same direction.

/ Highpass Filtering
x. wy F
original image sharpened image, 2I(7,c)-[(r,c-1)

11 October 2021 ©1999-2021 by Richard Alan Peters IT 25

YW The shift occurs because the EECE 4353 Image Processing
direction of the differencing Vanderbilt University School of Engineering
operation pushes edges in the

same direction. (see pp.7-8)

/ Highpass Filtering

sharpened image, 2I(r,c)-I(r,c-1) original image

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Apparent Shift due to HF Enhancement

original backward diff: I(r,c)-I(r,c-1) enhanced: 2I(r,c)-I(r,c-1)

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Apparent Shift due to HF Enhancement

enhanced: 21(7,c)-I(r,c-1) backward diff: I(r,c)-I(r,c-1) original

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Differentiation Through Integration

: [H#n](r40)=— [J Wen z-)h(p.z)dpaz

1 ae spel) Assume that h(p,x) =
2 2 ee
w=ax+By, Ja°+f? =1 &(p,x). Then I*h = I. and
By e | a(T+h)/ ow = aT/aw.
oO ‘ a, ¥ Differentiation property
9: 5| ae «| ~ jz { I(r ,c)} of the Fourier Transform.
= la? Pe 2/ew is a directional
z=au+ By, i ia derivative with direction
vector [a BJ" = [cos8,sin®].
H{ Th} =F {I} -F{h}
Convolution property of
I*h=5"'! {§ { i} ‘F { h}} the Fourier Transform.

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Differentiation Through Integration

s(ge}=s(t}s(h) (fess
=| j-3{1} |-5{h}

4. [a BJ" is a direction in
=5{1} | jz-5{h} | the plane. w and zare
projections along that
w=axt By, lo2 +f =1 direction.
= [2 Ds
eaaea is 4a ep =I The derivative of a
8 a convolution of I by h
5. —[Lfeh](7,c)= 1 Zar) is the convolution of I
ow ow by the derivative of h.

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Symmetric Differences

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21(r,c)—I(r,ce—1) 21(r,c)-—I(r—-1,c)
“wer ws10|

Al(r,c)—
I(r—1,c)—Mr + 1.c)—
I(r,e—1)—U(r,c +1)

11 October 2021

©1999-2021 by Richard Alan Peters II

31

an EECE 4353 Image Processing
Symmetric Differences Vanderbilt University School of Engineering

This computation
indexed over all the
rows and columns...

Al(r,c) —

I(r,c) I(r —1,c)—U(r+1,c)—
2] -I(r,c+1) UC+LOP ee )—1G,e-4

21(r,c)—I(r,e—1) 21(r,c)-—I(r—1,c)

-l “1
-l[2|-1 2 -1] 4 |-1
-l “I
t

.. is the same as
convolving the image
11 October 2021 ©1999-2021 by R with this kernel. 32

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A note on FBUD” differences & convolution

A backward difference on I is the same as ... ... right shifting a copy of I by one pixel
and subtracting it from I.

“forward, backward, up, down

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A note on FBUD” differences & convolution

That is, to compute
I(rg,¢)-1(r9,c-1) ...

.. convolve I with
h=[0 1 -1].

A backward difference is the same as... ... right shifting a copy and subtracting it.

—___—. h,=[0 1 -1] is a backward difference whereas
“forward, backward, up, down hy=[-11 0] is a forward difference.

11 October 2021 ©1999-2021 by Richard Alan Peters IT 34

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A note on FBUD” differences & convolution

backward diff: I(,c)-I(r,c-1) enhanced: 21(r,c)-I(r,c-1)

“forward, backward, up, down

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A note on FBUD” differences & convolution

enhanced: 21(,c)-I(r,c-1) original

“forward, backward, up, down

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Convolution Examples: Original Images

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a signed image; EECE 4353 Image Processing
Be Ois middle gray Vanderbilt University School of Engineering

Convolution Examples: Vertical Difference

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, signed image; EECE 4353 Image Processing
eet Ois middle gray Vanderbilt University School of Engineering

Convolution Examples: Horizontal Difference

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Oot G signed image; EECE 4353 Image Processing
ents Ois middle gray Vanderbilt University School of Engineering

Convolution Examples: H+ V_ Diff.

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w signed image; EECE 4353 Image Processing
A Ois middle gray Vanderbilt University School of Engineering

Convolution Examples: Diagonal Difference

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0-1 signed image: EECE 4353 Image Processing
oe Ois middle gray Vanderbilt University School of Engineering

Convolution Examples: Diagonal Difference

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0-1 signed image: EECE 4353 Image Processing
ne Ois middle gray Vanderbilt University School of Engineering

Convolution Examples: D+D Difference

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ees signed image; EECE 4353 Image Processing
ey Ois middle gray Vanderbilt University School of Engineering

Convolution Examples: H+ V+ D Diff.

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Convolution Examples: Original Images

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Original Image

power spectrum of I

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Left Difference

power spectrum of I*h = I*[-1 1 0] I+h = I*[0 1 -1]

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Original Image + Left Difference

H+(I+h) = +(I#[0 1 -1])

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Right Difference

power spectrum of I*h = I*[1 -1] T*h = I*[-1 1 0]

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image I +(I+h) = H(I+[-1 1 0])

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Vertical Edges (L+R Diffs)

power spectrum of I*h = I*[-1 2 -1]

Teh = I*[-1 2-1]

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image I H+(I*h) = H(I#[-1 2 -1])

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Down Difference

power spectrum of Ixh = T* [1] Ikh =I* [i i

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St

{EBSe2S8! i

image I T+(I+h) = 1+( I | )

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Up Difference

power spectrum of I*h = I* [| Ieh = T+{_j]

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EECE 4353 Image Processing

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en

ELL Lae
“Tart

image I Hh) = F+(1+|_j])

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Horizontal Edges (D+U Diffs)

power spectrum of Ixh = [* |

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©1999-2021 by Richard Alan Peters I 37

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a

ELLE

image I T+(i+h) = H+ | )

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Horiz. + Vert. tee (L+R+D+U Diffs)

power spectrum of I*h = I*

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original sharpened

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Original Image + Horiz. + Vert. Edges

—————
‘RISeESeES:

- F

sharpened original

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Unsharp Masking

is a film-photography darkroom technique for sharpening an image. A blurred copy of
the photonegative is contrast reduced and used to mask the original image.

original blurred negative

lo-contrast
° ate ° =

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Sharpening Through Blurring: Unsharp Masking

Let I be an image.
Let G, be a Gaussian convolution mask.

Then J =I * G, is a blurred image and K = I — J contains

all the high spatial frequencies from I.
| Often, the control, a, is

Define: | given as a percent value.
U=(lta) K+ J =a K+1L, | Then the formula is
where, typically 0<a <2. Pleuuaiss

U is called the unsharp masking of image I.

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Sharpening Through Blurring: Unsharp Masking

original image log power spectrum

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Sharpening Through Blurring: Unsharp Masking
>

Gaussian blur o=4 log power spectrum

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Sharpening Through Blurring: Unsharp Masking

original minus Gaussian blur log power spectrum

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A EECE 4353 Image Processing
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Sharpening Through Blurring: Unsharp Masking

unsharp masked image log power spectrum

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YW EECE 4353 Image Processing

Sharpening Through Blurring: Unsharp Masking

= 7

original image unsharp masked image

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Sharpening above a Specific Scale.

An image is sharpened by taking a linear combination of the
image and a highpass filtered version of itself. The scale of
the sharpening can be controlled via the cutoff of the HPF. In
the following examples the image has been sharpened via

Ineo =1+ Apts I+a(I [1*g(o)])=(1+a)I-a[1*2(o)],

where g is a 2D Gaussian with o € {1, 2, 4, 8, 16, 32, 64,
128, 256} and a is a scale factor, usually in (0,2). After the
computation, each image was histogram matched to I.

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7 EECE 4353 Image Processing
, 6)=0 Original Image Vanderbilt University School of Engineering

Sha EUNE above a Specific Scale

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YW oy =1, 0-1 EECE 4353 Image Processing

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ee above a Specific Seale

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YW y= 2, a=1 EECE 4353 Image Processing

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Sharpening above a Specific Scale

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W make EECE 4353 Image Processing

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see a aleve a Specific Scale

“esteunsan Nt 9
Petoly Thee

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YW y= 8, O= EECE 4353 Image Processing

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eDepenine above a Specific Scale

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YW = t6cesi EECE 4353 Image Processing

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_ Sees above a meade Scale

a

al} Desert: i eo) lai

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¥7 6 = 32, o=1 EECE 4353 Image Processing

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sbi! a above a Specific Seale

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YW oy = 64, a=1 EECE 4353 Image Processing

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pbapenine above a Specific Scale

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YW Gy = 128, a=! EECE 4353 Image Processing

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Sharpening above a Specific Scale

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7 Oy = 256, a=l EECE 4353 Image Processing

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Sharpening above a Specific Scale

= ae |

a

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wie EECE 4353 Image Processing
, 6)=0 Original Image Vanderbilt University School of Engineering

Sharpening above a Specific Scale

7 pies |

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YW EECE 4353 Image Processing

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Emphasizing a Specific Pass Band.

An image can be bandpass filtered by subtracting two differently
Gaussian filtered copies of it. That specific band can be
emphasized in the image by adding it back to the image. In the
following examples the image has been emphasized via

bpeay.o, I+ Ara, ray

-1+a[t-([tee(a)]-[lee(a)]]
=(1+a)I-a(I*[¢(o)-g(a))),

where oo, o, € {1, 2, 4, 8, 16, 32, 64, 128, 256} and a is a scale
factor, usually in (0,2). After the computation, each image was
histogram matched to I.

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YW (io= (0) 04 EECE 4353 Image Processing
pVo) = » a

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de a Specific Pass Band

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EECE 4353 Image Processing

, (91,09) = (2,1), a=1 Vanderbilt University School of Engineering

Beste a Specific Pass Band

Beaman cee |
etalyT es

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y (6,,0y) = (4,2), a=1 EECE 4353 Image Processing
B04 » A=

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Euiplasizine a Specific Pass Band -

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Y (6,,05) = (8,4), a=1 EECE 4353 Image Processing
b0'7 » A=

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Se a Specific Pass iiss

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y (61,09) = (16,8), o=1 EECE 4353 Image Processing
pO) = , A=

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ee a Specific Pass pe

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EECE 4353 Image Processing
, (01,59) = (32,16), a=1 Vanderbilt University School of Engineering

Emphasizing a Specific Pass Band

q ay

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EECE 4353 Image Processing
, (01,09) = (64,32), a=1 Vanderbilt University School of Engineering

Su a Specific Pass Band

f CIBC

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EECE 4353 Image Processing
, (01,09) = (128,64), a=1 Vanderbilt University School of Engineering

ee a Specific Pass Band
Sy

CBAC)

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EECE 4353 Image Processing
, (01,59) = (256,128), a=1 Vanderbilt University School of Engineering

Emphasizing a Specific Pass Band
sie 7

a

|

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eee EECE 4353 Image Processing
, 6)=0 Original Image Vanderbilt University School of Engineering

Emphasizing a Specific Pass Band

ae
Ly Petaly Tes
Va

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YW EECE 4353 Image Processing

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Noise Enhancement: the Problem with Sharpening

e Noise occurs in every natural imaging device
— Quantum effects in CCD arrays
— Random distribution of silver halide grains in film
— Neuronal noise in the retina

e Spatially independent noise
— The noise in one sensor has no effect on that in its neighbors
=> the autocorrelation of the signal is an impulse at the origin
— The chances of getting repeated patterns of any frequency are virtually nil
=> the frequency spectrum of the noise is flat

Recall: Autocorrelation = inverse Fourier transform of power
spectrum; Fourier transform of an impulse at (0,0) is a constant.

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Noise Enhancement: the Problem with Sharpening

The spectra of most natural images fall-off toward the high
frequencies.

IID noise has a flat spectrum.

Therefore, at some relatively high frequency (HF) the
energy in the noise is greater than that in the uncorrupted
image.

Sharpening multiplies the FT of the image by wu and v (or
by linear combinations of them) which, at HF, increases
the noise more than the uncorrupted image.

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YW Roy Sato, Dr. Aki Ross, Promo Stil from Final EECE 4353 Image Processing

Fantasy: The Spirits Within, Square Pictures, 2001. Vanderbilt University School of Engineering

Effects of Noise on Linear Enhancement of HF

original image noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 94

A Roy Sato, Dr. Aki Ross, Promo Stil from Final EECE 4353 Image Processing

Fantasy: The Spirits Within, Square Pictures, 2001, Vanderbilt University School of Engineering

Effects of Noise on Linear Enhancement of HF

HF enhanced original HF enhanced noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 95

Commercial Product: : i ‘
Topaz Labs Sharpen Vanderbilt University School of Engineering

YW renee EECE 4353 Image Processing

Effects of Noise on Linear Enhancement of HF

Topaz HF enhanced original Topaz HF enhanced noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 96

YW Roy Sato, Dr. Aki Ross, Promo Stil from Final EECE 4353 Image Processing

Fantasy: The Spirits Within, Square Pictures, 2001. Vanderbilt University School of Engineering

Effects of Noise on Linear Enhancement of HF

original image noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 97

YW Roy Sato, Dr. Aki Ross, Promo Stil from Final EECE 4353 Image Processing

Fantasy: The Spirits Within, Square Pictures, 2001. Vanderbilt University School of Engineering

Effects of Noise on Linear Enhancement of HF

original image noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 98

A Roy Sato, Dr. Aki Ross, Promo Stil from Final EECE 4353 Image Processing

Fantasy: The Spirits Within, Square Pictures, 2001, Vanderbilt University School of Engineering

Effects of Noise on Linear Enhancement of HF

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HF enhanced original HF enhanced noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 99

Commercial Product: ‘ ‘ "
Topaz Labs Sharpen Vanderbilt University School of Engineering

Y pre ena iee EECE 4353 Image Processing

Effects of Noise on Linear Enhancement of HF

Topaz HF enhanced original Topaz HF enhanced noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 100

YW Roy Sato, Dr. Aki Ross, Promo Stil from Final EECE 4353 Image Processing

Fantasy: The Spirits Within, Square Pictures, 2001. Vanderbilt University School of Engineering

Effects of Noise on Linear Enhancement of HF

original image noisy image

11 October 2021 ©1999-2021 by Richard Alan Peters IT 101

EECE 4353 Image Processing

Vanderbilt University School of Engineering

AI/N eural Network Models = Image shaypenms

Tutorial: Deep Learning based Super Resolution

11 October 2021 ©1999-2021 by Richard Alan Peters IT 102